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(f)    Let (An) be the sequence of eigenvalues of a nuclear operator Ty each counted
according to its algebraic multiplicity (11.4.1). Show that]T \Xn\ rg \\T\\i, (For each


integer^, consider the sum V of the p spaces N(JU*; T) (I <J k <*p), where filt..., fj.p
are the first p distinct eigenvalues in the sequence (An). Take a Hilbert basis of
V with respect to which the matrix of T\ V is triangular, and use (c).) Deduce that if
T is a Hilbert-Schmidt operator and if (An) is the sequence of its eigenvalues, each
counted with its algebraic multiplicity, then]T |An|2 <£ \\T\\Z-


(g)    If a continuous operator Te J*?(E) is such that, for each pair of Hilbert
bases (an\ (bn) of E, the series £ (T * an \ bn) is convergent, then T is nuclear. (Write


T = LU* (Problem 6(a)), and by choosing (an) and (bn) suitably show that Llf2 is a
Hilbert-Schmidt operator.)

(h) For a continuous operator Te &(E) to be nuclear, it is necessary and sufficient
that, for at least one Hilbert basis (<?„) of E, the seriesX \\T- en\\ should converge.

(Write T= UR (Problem 6(a)) and note that (R-en\ en) <; || T - en\\. This proves that
the condition is sufficient. Conversely, take for (en) a basis consisting of eigenvectors
of R.)

(i) In the space E = /£, let (en) be the canonical Hilbert basis, and let
a —1L OA*Xi- If F is the subspace C • a of dimension 1, show that the projector PF


is nuclear, but that the series]T \\PF - en\\ does not converge.


8.    Show that, for every normal operator W on a separable Hilbert space E, there exists
a normal operator N' such that N'2 = TV. Give examples where there exist infinitely
many such operators,

9.    Let T be a continuous operator on a separable Hilbert space E.

(a)    For T to be a topological left zero-divisor (Section 15.2, Problem 3) in the algebra
^f(E), it is necessary and sufficient that there should exist a sequence (xn) of vectors
in E such that \\xn\\ = 1 for all n and such that (T • xn) tends to 0. The complex numbers
£ such that T— £ • 1 is a topological left zero-divisor in ^(E) form what is called the
approximative point-spectrum Spa(T). Thus £ $ Spa(3n) means that T— £ • 1 is injective
and a horneomorphism of E onto a closed subspace of E. Show that Spa(T) is closed
in C, and contains the frontier of Sp(!T). If P is any polynomial, show that


(b)    LetTe JS?(E). For each A e C, let m(T, A) denote the dimension of Ker(r*- A • 1)
(so that m(T, A) is either a nonnegative integer or + oo), equal also to the codimension of
(T— A • 1)(E), Let r0 e £?(E) and let A0 be a complex number not belonging to
Spa(To). Show that there exists a number e>0 such that m(T, A) = m(TQ, A0)
whenever \\T- TQ\\ <; e and |A - A0| g e.

(c)    Deduce from (b) that if K is a compact subset of C which does not intersect
Spa(J'o), then there exists e>0 such that w(rğA)— m(r0,A) for all AeK and all
Te &(E) such that \\T-TQ\\<: e.

(d)    Let T e &(E) and let K be a compact subset of Sp(7) - Spa(T) with the follow-
ing properties: (1) 0 $K; (2) the inverse image K/ of K under the mapping £i—ğ£2
is convex; (3) w(r,A)= 1 for all AeK. Then there exists no operator T'e &(E)
such that T/2—T, (Supposing the contrary, let L = K/nSp(r/); then we have
L <= Sp(r) - Spa(TO, and L u (-L) = K'. Show that A e L implies that ~A f Low that the function